The heat kernel of the Laplacian defined on a uniform grid

We adapt a method originally developed by E.B. Davies for second order elliptic operators to obtain an upper heat kernel bound for the Laplacian defined on a uniform grid on the plane.     

The Norm of the Laplace Operator Restriction to a Homogeneous Polynomial Space

We investigate the restriction Δ _r,μ of the Laplace operator Δ onto the space of r-variate homogeneous polynomials F of degree μ. In the uniform norm on the unit ball of ℝ ^r , and with the corresponding operator norm, ‖Δ _r,μ F‖≤‖Δ _r,μ ‖⋅‖F‖ holds, where, for arbitrary F, the ‘constant’ ‖Δ _r,μ ‖ is the best possible. We describe ‖Δ _r,μ ‖ with the help of the family T _μ (σ x), , of scaled Chebyshev polynomials of degree μ. On the interval [−1,+1], they alternate at least (μ−1)-times, as the Zolotarev polynomials do, but they differ from them by their symmetry. We call them Zolotarev polynomials of the second kind, and calculate ‖Δ _r,μ ‖ with their help. We derive upper and lower bounds, as well as the asymptotics for μ→∞. For r≥5 and sufficiently large μ, we just get ‖Δ _r,μ ‖=(r−2)μ(μ−1). However, for 2≤r≤4 or lower values of μ, the result is more complicated. This gives the problem a particular flavor. Some Bessel functions and the φcot φ-expansion are involved.      

The Real Powers of the Convolution of a Gamma Distribution and a Bernoulli Distribution

In this paper, we essentially compute the set of x,y>0 such that the mapping \(z\longmapsto(1-r+re^{z})^{x}(\frac{\lambda}{\lambda-z})^{y}\) is a Laplace transform. If X and Y are two independent random variables which have respectively Bernoulli and Gamma distributions, we denote by μ the distribution of X+Y. The above problem is equivalent to finding the set of x>0 such that μ ^*x exists.     

The effective application of a new approach to the generalized orienteering problem

The Orienteering Problem (OP) is an important problem in network optimization in which each city in a network is assigned a score and a maximum-score path from a designated start city to a designated end city is sought that is shorter than a pre-specified length limit. The Generalized Orienteering Problem (GOP) is a generalized version of the OP in which each city is assigned a number of scores for different attributes and the overall function to optimize is a function of these attribute scores. In this paper, the function used was a non-linear combination of attribute scores, making the problem difficult to solve. The GOP has a number of applications, largely in the field of routing. We designed a two-parameter iterative algorithm for the GOP, and computational experiments suggest that this algorithm performs as well as or better than other heuristics for the GOP in terms of solution quality while running faster. Further computational experiments suggest that our algorithm also outperforms the leading algorithm for solving the OP in terms of solution quality while maintaining a comparable solution speed.     

The Poincaré series of divisorial valuations in the plane defines the topology of the set of divisors

With a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincaré series of this filtration turns out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincaré series of the corresponding multi-index filtration similar to the one associated with plane germs. Here we show that the Poincaré series of a set of divisorial valuations on the ring of germs of functions of two variables defines “the topology of the set of the divisors” in the sense that it defines the minimal resolution of this set up to combinatorial equivalence. For the plane curve singularity case, we also give a somewhat simpler proof of the statement by Yamamoto which shows that the Alexander polynomial is equivalent to the embedded topology.     

The Ratio of the Perimeter and the Area of Unions of Copies of a Fixed Set

T. Keleti asked 10 years ago whether the ratio of the perimeter and the area of a union of unit squares is always at most 4. In the present paper we show that the ratio is less than 5.6.     

An Analytical Study of Boundary Layer Flows on a Continuous Stretching Surface

In this paper the momentum and heat transfer characteristics for a self-similarity boundary layer on exponentially stretching surface modeled by a system of nonlinear differential equations is studied. The system is solved using the Homotopy Analysis Method (HAM), which yields an analytic solution in the form of a rapidly convergent infinite series with easily computable terms. Homotopy analysis method contains the auxiliary parameter ℏ, which provides us with a simple way to adjust and control the convergence region of solution series. By suitable choice of the auxiliary parameter ℏ, reasonable solutions for large modulus can be obtained.     

On the order of a non-abelian representation group of a slim dense near hexagon

In this paper we study the possible orders of a non-abelian representation group of a slim dense near hexagon. We prove that if the representation group R of a slim dense near hexagon S is non-abelian, then R is a 2-group of exponent 4 and |R|=2 ^β , 1+NPdim(S)≤β≤1+dimV(S), where NPdim(S) is the near polygon embedding dimension of S and dimV(S) is the dimension of the universal representation module V(S) of S. Further, if β=1+NPdim(S), then R is necessarily an extraspecial 2-group. In that case, we determine the type of the extraspecial 2-group in each case. We also deduce that the universal representation group of S is a central product of an extraspecial 2-group and an abelian 2-group of exponent at most 4. This work was partially done when B.K. Sahoo was a Research Fellow at the Indian Statistical Institute, Bangalore Center with NBHM fellowship, DAE Grant 39/3/2000-R&D-II, Govt. of India.     

The Method of Virtual Components in the Multivariate Setting

We describe the so-called method of virtual components for tight wavelet framelets to increase their approximation order and vanishing moments in the multivariate setting. Two examples of the virtual components for tight wavelet frames based on bivariate box splines on three or four direction mesh are given. As a byproduct, a new construction of tight wavelet frames based on box splines under the quincunx dilation matrix is presented.     

The topological center of the spectrum of some distal algebras

The topological center of the spectrum of the Weyl algebra W, i.e. the norm closure of the algebra generated by the set of functions , is characterized in a recent paper by Jabbari and Namioka (Ellis group and the topological center of the flow generated by the map , to appear in Milan J. Math.). By the techniques essentially used in the cited paper, the topological center of the spectrum of the subalgebra W _k , the norm closure of the algebra generated by the set of functions , will be characterized, for all k∈ℕ. Also an example of a non-minimal dynamical system, with the enveloping semigroup Σ, for which the set of all continuous elements of Σ is not equal to the topological center of Σ, is given.     

On the Differential Geometric Characterization of The Lee Models

This paper pertains to the J-Hermitian geometry of model domains introduced by Lee (Mich. Math. J. 54(1), 179–206, 2006; J. Reine Angew. Math. 623, 123–160, 2008). We first construct a Hermitian invariant metric on the Lee model and show that the invariant metric actually coincides with the Kobayashi-Royden metric, thus demonstrating an uncommon phenomenon that the Kobayashi-Royden metric is J-Hermitian in this case. Then we follow Cartan’s differential-form approach and find differential-geometric invariants, including torsion invariants, of the Lee model equipped with this J-Hermitian Kobayashi-Royden metric, and present a theorem that characterizes the Lee model by those invariants, up to J-holomorphic isometric equivalence. We also present an all dimensional analysis of the asymptotic behavior of the Kobayashi metric near the strongly pseudoconvex boundary points of domains in almost complex manifolds.     

The Hausdorff Dimension of the Double Points on the Brownian Frontier

The frontier of a planar Brownian motion is the boundary of the unbounded component of the complement of its range. In this paper, we find the Hausdorff dimension of the set of double points on the frontier.     

Equivalent Conditions of Asymptotics for the Density of the Supremum of a Random Walk in the Intermediate Case

This paper obtains some equivalent conditions about the asymptotics for the density of the supremum of a random walk with light-tailed increments in the intermediate case. To do this, the paper first corrects the proofs of some existing results about densities of random sums. On the basis of the above results, the paper obtains some equivalent conditions about the asymptotics for densities of ruin distributions in the intermediate case and densities of infinitely divisible distributions. In the above studies, some differences and relations between the results on a distribution and its corresponding density can be discovered.      

The Euclidean Distortion of the Lamplighter Group

We show that the cyclic lamplighter group C ₂ ? C _n embeds into Hilbert space with distortion $\mathrm{O}(\sqrt{\log n})We show that the cyclic lamplighter group C ₂ ≀ C _n embeds into Hilbert space with distortion O(?{logn})\mathrm{O}(\sqrt{\log n}) . This matches the lower bound proved by Lee et al. (Geom. Funct. Anal., 2009), answering a question posed in that paper. Thus, the Euclidean distortion of C ₂ ≀ C _n is \varTheta(?{logn})\varTheta(\sqrt{\log n}) . Our embedding is constructed explicitly in terms of the irreducible representations of the group. Since the optimal Euclidean embedding of a finite group can always be chosen to be equivariant, as shown by Aharoni et al. (Isr. J. Math. 52(3):251–265, 1985) and by Gromov (see de Cornulier et. al. in Geom. Funct. Anal., 2009), such representation-theoretic considerations suggest a general tool for obtaining upper and lower bounds on Euclidean embeddings of finite groups.     

Differentiability of Intrinsic Lipschitz Functions within Heisenberg Groups

We study the notion of intrinsic Lipschitz graphs within Heisenberg groups, focusing our attention on their Hausdorff dimension and on the almost everywhere existence of (geometrically defined) tangent subgroups. In particular, a Rademacher type theorem enables us to prove that previous definitions of rectifiable sets in Heisenberg groups are natural ones.     

The First Exit Time of a Brownian Motion from the Minimum and Maximum Parabolic Domains

Consider a Brownian motion starting at an interior point of the minimum or maximum parabolic domains, namely, D_min={(x,y₁,y₂):||x|| < min{(y₁+1)^1/p₁,(y₂+1)^1/p₂}}D_{\min}=\{(x,y_{1},y_{2}):\|x\|< \min\{(y_{1}+1)^{1/p_{1}},(y_{2}+1)^{1/p_{2}}\}\} and D_max={(x,y₁,y₂):||x|| < max{(y₁+1)^1/p₁,\allowbreak(y₂+1)^1/p₂}}D_{\max}=\{(x,y_{1},y_{2}):\|x\|<\max\{(y_{1}+1)^{1/p_{1}},\allowbreak(y_{2}+1)^{1/p_{2}}\}\} in R ^d+2,d≥1, respectively, where ‖⋅‖ is the Euclidean norm in R ^d , y ₁,y ₂≥−1, and p ₁,p ₂>1. Let t_Dmin\tau_{D_{\min}} and t_Dmax\tau_{D_{\max}} denote the first times the Brownian motion exits from D _min and D _max. Estimates with exact constants for the asymptotics of $\log P(\tau_{D_{\min}}>t)$\log P(\tau_{D_{\min}}>t) and $\log P(\tau_{D_{\max}}>t)$\log P(\tau_{D_{\max}}>t) are given as t→∞, depending on the relationship between p ₁ and p ₂, respectively. The proof methods are based on Gordon’s inequality and early works of Li, Lifshits, and Shi in the single general parabolic domain case.     

The Overlay of Minimization Diagrams in a Randomized Incremental Construction

In a randomized incremental construction of the minimization diagram of a collection of n hyperplanes in ℝ ^d , for d≥2, the hyperplanes are inserted one by one, in a random order, and the minimization diagram is updated after each insertion. We show that if we retain all the versions of the diagram, without removing any old feature that is now replaced by new features, the expected combinatorial complexity of the resulting overlay does not grow significantly. Specifically, this complexity is O(n ^⌊d/2⌋log n), for d odd, and O(n ^⌊d/2⌋), for d even. The bound is asymptotically tight in the worst case for d even, and we show that this is also the case for d=3. Several implications of this bound, mainly its relation to approximate halfspace range counting, are also discussed.     

The group of symmetries of the shorter Moonshine module

It is shown that the automorphism group of the shorter Moonshine module VB ^♮ constructed in the author’s Ph.D. thesis (Selbstduale Vertexoperatorsuperalgebren und das Babymonster, Bonner Math. Schr. 286: 1996) is the direct product of the finite simple group known as the Baby Monster and the cyclic group of order 2.     

A Note on the Distribution of the Distance from a Lattice

Let ℒ be an n-dimensional lattice, and let x be a point chosen uniformly from a large ball in ℝ ⁿ . In this note we consider the distribution of the distance from x to ℒ, normalized by the largest possible such distance (i.e., the covering radius of ℒ). By definition, the support of this distribution is [0,1]. We show that there exists a universal constant α ₂ that provides a natural “threshold” for this distribution in the following sense. For any ε>0, there exists a δ>0 such that for any lattice, this distribution has mass at least δ on [α ₂−ε,1]; moreover, there exist lattices for which the distribution is tightly concentrated around α ₂ (and so the mass on [α ₂+ε,1] can be arbitrarily small). We also provide several bounds on α ₂ and its extension to other ℓ _p norms. We end with an application from the area of computational complexity. Namely, we show that α ₂ is exactly the approximation factor of a certain natural protocol for the Covering Radius Problem. I. Haviv’s research was supported by the Binational Science Foundation and by the Israel Science Foundation. V. Lyubashevsky’s research was supported by NSF ITR 0313241. O. Regev’s research was supported by an Alon Fellowship, by the Binational Science Foundation, by the Israel Science Foundation, and by the European Commission under the Integrated Project QAP funded by the IST directorate as Contract Number 015848.     

On the number of divisors which are values of a polynomial

Let τ(n) be the number of positive divisors of an integer n, and for a polynomial P(X)∈ℤ[X], let